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Theorem pw2dvdslemn 10253
Description: Lemma for pw2dvds 10254. If a natural number has some power of two which does not divide it, there is a highest power of two which does divide it. (Contributed by Jim Kingdon, 14-Nov-2021.)
Assertion
Ref Expression
pw2dvdslemn  |-  ( ( N  e.  NN  /\  A  e.  NN  /\  -.  ( 2 ^ A
)  ||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) )
Distinct variable group:    m, N
Allowed substitution hint:    A( m)

Proof of Theorem pw2dvdslemn
Dummy variables  w  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 3simpb 913 . 2  |-  ( ( N  e.  NN  /\  A  e.  NN  /\  -.  ( 2 ^ A
)  ||  N )  ->  ( N  e.  NN  /\ 
-.  ( 2 ^ A )  ||  N
) )
2 oveq2 5548 . . . . . . . 8  |-  ( w  =  1  ->  (
2 ^ w )  =  ( 2 ^ 1 ) )
32breq1d 3802 . . . . . . 7  |-  ( w  =  1  ->  (
( 2 ^ w
)  ||  N  <->  ( 2 ^ 1 )  ||  N ) )
43notbid 602 . . . . . 6  |-  ( w  =  1  ->  ( -.  ( 2 ^ w
)  ||  N  <->  -.  (
2 ^ 1 ) 
||  N ) )
54anbi2d 445 . . . . 5  |-  ( w  =  1  ->  (
( N  e.  NN  /\ 
-.  ( 2 ^ w )  ||  N
)  <->  ( N  e.  NN  /\  -.  (
2 ^ 1 ) 
||  N ) ) )
65imbi1d 224 . . . 4  |-  ( w  =  1  ->  (
( ( N  e.  NN  /\  -.  (
2 ^ w ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) )  <->  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) ) )
7 oveq2 5548 . . . . . . . 8  |-  ( w  =  k  ->  (
2 ^ w )  =  ( 2 ^ k ) )
87breq1d 3802 . . . . . . 7  |-  ( w  =  k  ->  (
( 2 ^ w
)  ||  N  <->  ( 2 ^ k )  ||  N ) )
98notbid 602 . . . . . 6  |-  ( w  =  k  ->  ( -.  ( 2 ^ w
)  ||  N  <->  -.  (
2 ^ k ) 
||  N ) )
109anbi2d 445 . . . . 5  |-  ( w  =  k  ->  (
( N  e.  NN  /\ 
-.  ( 2 ^ w )  ||  N
)  <->  ( N  e.  NN  /\  -.  (
2 ^ k ) 
||  N ) ) )
1110imbi1d 224 . . . 4  |-  ( w  =  k  ->  (
( ( N  e.  NN  /\  -.  (
2 ^ w ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) )  <->  ( ( N  e.  NN  /\  -.  ( 2 ^ k
)  ||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) ) )
12 oveq2 5548 . . . . . . . 8  |-  ( w  =  ( k  +  1 )  ->  (
2 ^ w )  =  ( 2 ^ ( k  +  1 ) ) )
1312breq1d 3802 . . . . . . 7  |-  ( w  =  ( k  +  1 )  ->  (
( 2 ^ w
)  ||  N  <->  ( 2 ^ ( k  +  1 ) )  ||  N ) )
1413notbid 602 . . . . . 6  |-  ( w  =  ( k  +  1 )  ->  ( -.  ( 2 ^ w
)  ||  N  <->  -.  (
2 ^ ( k  +  1 ) ) 
||  N ) )
1514anbi2d 445 . . . . 5  |-  ( w  =  ( k  +  1 )  ->  (
( N  e.  NN  /\ 
-.  ( 2 ^ w )  ||  N
)  <->  ( N  e.  NN  /\  -.  (
2 ^ ( k  +  1 ) ) 
||  N ) ) )
1615imbi1d 224 . . . 4  |-  ( w  =  ( k  +  1 )  ->  (
( ( N  e.  NN  /\  -.  (
2 ^ w ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) )  <->  ( ( N  e.  NN  /\  -.  ( 2 ^ (
k  +  1 ) )  ||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) ) )
17 oveq2 5548 . . . . . . . 8  |-  ( w  =  A  ->  (
2 ^ w )  =  ( 2 ^ A ) )
1817breq1d 3802 . . . . . . 7  |-  ( w  =  A  ->  (
( 2 ^ w
)  ||  N  <->  ( 2 ^ A )  ||  N ) )
1918notbid 602 . . . . . 6  |-  ( w  =  A  ->  ( -.  ( 2 ^ w
)  ||  N  <->  -.  (
2 ^ A ) 
||  N ) )
2019anbi2d 445 . . . . 5  |-  ( w  =  A  ->  (
( N  e.  NN  /\ 
-.  ( 2 ^ w )  ||  N
)  <->  ( N  e.  NN  /\  -.  (
2 ^ A ) 
||  N ) ) )
2120imbi1d 224 . . . 4  |-  ( w  =  A  ->  (
( ( N  e.  NN  /\  -.  (
2 ^ w ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) )  <->  ( ( N  e.  NN  /\  -.  ( 2 ^ A
)  ||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) ) )
22 0nn0 8254 . . . . . 6  |-  0  e.  NN0
2322a1i 9 . . . . 5  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  0  e.  NN0 )
24 oveq2 5548 . . . . . . . 8  |-  ( m  =  0  ->  (
2 ^ m )  =  ( 2 ^ 0 ) )
2524breq1d 3802 . . . . . . 7  |-  ( m  =  0  ->  (
( 2 ^ m
)  ||  N  <->  ( 2 ^ 0 )  ||  N ) )
26 oveq1 5547 . . . . . . . . . 10  |-  ( m  =  0  ->  (
m  +  1 )  =  ( 0  +  1 ) )
2726oveq2d 5556 . . . . . . . . 9  |-  ( m  =  0  ->  (
2 ^ ( m  +  1 ) )  =  ( 2 ^ ( 0  +  1 ) ) )
2827breq1d 3802 . . . . . . . 8  |-  ( m  =  0  ->  (
( 2 ^ (
m  +  1 ) )  ||  N  <->  ( 2 ^ ( 0  +  1 ) )  ||  N ) )
2928notbid 602 . . . . . . 7  |-  ( m  =  0  ->  ( -.  ( 2 ^ (
m  +  1 ) )  ||  N  <->  -.  (
2 ^ ( 0  +  1 ) ) 
||  N ) )
3025, 29anbi12d 450 . . . . . 6  |-  ( m  =  0  ->  (
( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
)  <->  ( ( 2 ^ 0 )  ||  N  /\  -.  ( 2 ^ ( 0  +  1 ) )  ||  N ) ) )
3130adantl 266 . . . . 5  |-  ( ( ( N  e.  NN  /\ 
-.  ( 2 ^ 1 )  ||  N
)  /\  m  = 
0 )  ->  (
( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
)  <->  ( ( 2 ^ 0 )  ||  N  /\  -.  ( 2 ^ ( 0  +  1 ) )  ||  N ) ) )
32 2cnd 8063 . . . . . . . 8  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  2  e.  CC )
3332exp0d 9543 . . . . . . 7  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  ( 2 ^ 0 )  =  1 )
34 simpl 106 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  N  e.  NN )
3534nnzd 8418 . . . . . . . 8  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  N  e.  ZZ )
36 1dvds 10122 . . . . . . . 8  |-  ( N  e.  ZZ  ->  1  ||  N )
3735, 36syl 14 . . . . . . 7  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  1  ||  N
)
3833, 37eqbrtrd 3812 . . . . . 6  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  ( 2 ^ 0 )  ||  N
)
39 simpr 107 . . . . . . 7  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  -.  ( 2 ^ 1 )  ||  N )
40 0p1e1 8104 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
4140oveq2i 5551 . . . . . . . 8  |-  ( 2 ^ ( 0  +  1 ) )  =  ( 2 ^ 1 )
4241breq1i 3799 . . . . . . 7  |-  ( ( 2 ^ ( 0  +  1 ) ) 
||  N  <->  ( 2 ^ 1 )  ||  N )
4339, 42sylnibr 612 . . . . . 6  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  -.  ( 2 ^ ( 0  +  1 ) )  ||  N )
4438, 43jca 294 . . . . 5  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  ( ( 2 ^ 0 )  ||  N  /\  -.  ( 2 ^ ( 0  +  1 ) )  ||  N ) )
4523, 31, 44rspcedvd 2680 . . . 4  |-  ( ( N  e.  NN  /\  -.  ( 2 ^ 1 )  ||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) )
46 simpll 489 . . . . . . . . 9  |-  ( ( ( k  e.  NN  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  (
2 ^ k ) 
||  N )  -> 
k  e.  NN )
4746nnnn0d 8292 . . . . . . . 8  |-  ( ( ( k  e.  NN  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  (
2 ^ k ) 
||  N )  -> 
k  e.  NN0 )
48 oveq2 5548 . . . . . . . . . . 11  |-  ( m  =  k  ->  (
2 ^ m )  =  ( 2 ^ k ) )
4948breq1d 3802 . . . . . . . . . 10  |-  ( m  =  k  ->  (
( 2 ^ m
)  ||  N  <->  ( 2 ^ k )  ||  N ) )
50 oveq1 5547 . . . . . . . . . . . . 13  |-  ( m  =  k  ->  (
m  +  1 )  =  ( k  +  1 ) )
5150oveq2d 5556 . . . . . . . . . . . 12  |-  ( m  =  k  ->  (
2 ^ ( m  +  1 ) )  =  ( 2 ^ ( k  +  1 ) ) )
5251breq1d 3802 . . . . . . . . . . 11  |-  ( m  =  k  ->  (
( 2 ^ (
m  +  1 ) )  ||  N  <->  ( 2 ^ ( k  +  1 ) )  ||  N ) )
5352notbid 602 . . . . . . . . . 10  |-  ( m  =  k  ->  ( -.  ( 2 ^ (
m  +  1 ) )  ||  N  <->  -.  (
2 ^ ( k  +  1 ) ) 
||  N ) )
5449, 53anbi12d 450 . . . . . . . . 9  |-  ( m  =  k  ->  (
( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
)  <->  ( ( 2 ^ k )  ||  N  /\  -.  ( 2 ^ ( k  +  1 ) )  ||  N ) ) )
5554adantl 266 . . . . . . . 8  |-  ( ( ( ( k  e.  NN  /\  ( N  e.  NN  /\  -.  ( 2 ^ (
k  +  1 ) )  ||  N ) )  /\  ( 2 ^ k )  ||  N )  /\  m  =  k )  -> 
( ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  + 
1 ) )  ||  N )  <->  ( (
2 ^ k ) 
||  N  /\  -.  ( 2 ^ (
k  +  1 ) )  ||  N ) ) )
56 simpr 107 . . . . . . . . 9  |-  ( ( ( k  e.  NN  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  (
2 ^ k ) 
||  N )  -> 
( 2 ^ k
)  ||  N )
57 simplrr 496 . . . . . . . . 9  |-  ( ( ( k  e.  NN  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  (
2 ^ k ) 
||  N )  ->  -.  ( 2 ^ (
k  +  1 ) )  ||  N )
5856, 57jca 294 . . . . . . . 8  |-  ( ( ( k  e.  NN  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  (
2 ^ k ) 
||  N )  -> 
( ( 2 ^ k )  ||  N  /\  -.  ( 2 ^ ( k  +  1 ) )  ||  N
) )
5947, 55, 58rspcedvd 2680 . . . . . . 7  |-  ( ( ( k  e.  NN  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  (
2 ^ k ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) )
6059adantllr 458 . . . . . 6  |-  ( ( ( ( k  e.  NN  /\  ( ( N  e.  NN  /\  -.  ( 2 ^ k
)  ||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) )  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  (
2 ^ k ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) )
61 simprl 491 . . . . . . . 8  |-  ( ( ( k  e.  NN  /\  ( ( N  e.  NN  /\  -.  (
2 ^ k ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) )  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  ->  N  e.  NN )
6261anim1i 327 . . . . . . 7  |-  ( ( ( ( k  e.  NN  /\  ( ( N  e.  NN  /\  -.  ( 2 ^ k
)  ||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) )  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  -.  ( 2 ^ k
)  ||  N )  ->  ( N  e.  NN  /\ 
-.  ( 2 ^ k )  ||  N
) )
63 simpllr 494 . . . . . . 7  |-  ( ( ( ( k  e.  NN  /\  ( ( N  e.  NN  /\  -.  ( 2 ^ k
)  ||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) )  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  -.  ( 2 ^ k
)  ||  N )  ->  ( ( N  e.  NN  /\  -.  (
2 ^ k ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) )
6462, 63mpd 13 . . . . . 6  |-  ( ( ( ( k  e.  NN  /\  ( ( N  e.  NN  /\  -.  ( 2 ^ k
)  ||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) )  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  /\  -.  ( 2 ^ k
)  ||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) )
65 2nn 8144 . . . . . . . . 9  |-  2  e.  NN
66 simpll 489 . . . . . . . . . 10  |-  ( ( ( k  e.  NN  /\  ( ( N  e.  NN  /\  -.  (
2 ^ k ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) )  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  ->  k  e.  NN )
6766nnnn0d 8292 . . . . . . . . 9  |-  ( ( ( k  e.  NN  /\  ( ( N  e.  NN  /\  -.  (
2 ^ k ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) )  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  ->  k  e.  NN0 )
68 nnexpcl 9433 . . . . . . . . 9  |-  ( ( 2  e.  NN  /\  k  e.  NN0 )  -> 
( 2 ^ k
)  e.  NN )
6965, 67, 68sylancr 399 . . . . . . . 8  |-  ( ( ( k  e.  NN  /\  ( ( N  e.  NN  /\  -.  (
2 ^ k ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) )  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  ->  (
2 ^ k )  e.  NN )
7061nnzd 8418 . . . . . . . 8  |-  ( ( ( k  e.  NN  /\  ( ( N  e.  NN  /\  -.  (
2 ^ k ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) )  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  ->  N  e.  ZZ )
71 dvdsdc 10116 . . . . . . . 8  |-  ( ( ( 2 ^ k
)  e.  NN  /\  N  e.  ZZ )  -> DECID  ( 2 ^ k ) 
||  N )
7269, 70, 71syl2anc 397 . . . . . . 7  |-  ( ( ( k  e.  NN  /\  ( ( N  e.  NN  /\  -.  (
2 ^ k ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) )  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  -> DECID  ( 2 ^ k
)  ||  N )
73 exmiddc 755 . . . . . . 7  |-  (DECID  ( 2 ^ k )  ||  N  ->  ( ( 2 ^ k )  ||  N  \/  -.  (
2 ^ k ) 
||  N ) )
7472, 73syl 14 . . . . . 6  |-  ( ( ( k  e.  NN  /\  ( ( N  e.  NN  /\  -.  (
2 ^ k ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) )  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  ->  (
( 2 ^ k
)  ||  N  \/  -.  ( 2 ^ k
)  ||  N )
)
7560, 64, 74mpjaodan 722 . . . . 5  |-  ( ( ( k  e.  NN  /\  ( ( N  e.  NN  /\  -.  (
2 ^ k ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) )  /\  ( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
) )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  + 
1 ) )  ||  N ) )
7675exp31 350 . . . 4  |-  ( k  e.  NN  ->  (
( ( N  e.  NN  /\  -.  (
2 ^ k ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) )  ->  (
( N  e.  NN  /\ 
-.  ( 2 ^ ( k  +  1 ) )  ||  N
)  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  + 
1 ) )  ||  N ) ) ) )
776, 11, 16, 21, 45, 76nnind 8006 . . 3  |-  ( A  e.  NN  ->  (
( N  e.  NN  /\ 
-.  ( 2 ^ A )  ||  N
)  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  + 
1 ) )  ||  N ) ) )
78773ad2ant2 937 . 2  |-  ( ( N  e.  NN  /\  A  e.  NN  /\  -.  ( 2 ^ A
)  ||  N )  ->  ( ( N  e.  NN  /\  -.  (
2 ^ A ) 
||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) ) )
791, 78mpd 13 1  |-  ( ( N  e.  NN  /\  A  e.  NN  /\  -.  ( 2 ^ A
)  ||  N )  ->  E. m  e.  NN0  ( ( 2 ^ m )  ||  N  /\  -.  ( 2 ^ ( m  +  1 ) )  ||  N
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    <-> wb 102    \/ wo 639  DECID wdc 753    /\ w3a 896    = wceq 1259    e. wcel 1409   E.wrex 2324   class class class wbr 3792  (class class class)co 5540   0cc0 6947   1c1 6948    + caddc 6950   NNcn 7990   2c2 8040   NN0cn0 8239   ZZcz 8302   ^cexp 9419    || cdvds 10108
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-mulrcl 7041  ax-addcom 7042  ax-mulcom 7043  ax-addass 7044  ax-mulass 7045  ax-distr 7046  ax-i2m1 7047  ax-1rid 7049  ax-0id 7050  ax-rnegex 7051  ax-precex 7052  ax-cnre 7053  ax-pre-ltirr 7054  ax-pre-ltwlin 7055  ax-pre-lttrn 7056  ax-pre-apti 7057  ax-pre-ltadd 7058  ax-pre-mulgt0 7059  ax-pre-mulext 7060  ax-arch 7061
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-reu 2330  df-rmo 2331  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-if 3360  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-frec 6009  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-i1p 6623  df-iplp 6624  df-iltp 6626  df-enr 6869  df-nr 6870  df-ltr 6873  df-0r 6874  df-1r 6875  df-0 6954  df-1 6955  df-r 6957  df-lt 6960  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125  df-sub 7247  df-neg 7248  df-reap 7640  df-ap 7647  df-div 7726  df-inn 7991  df-2 8049  df-n0 8240  df-z 8303  df-uz 8570  df-q 8652  df-rp 8682  df-fl 9222  df-mod 9273  df-iseq 9376  df-iexp 9420  df-dvds 10109
This theorem is referenced by:  pw2dvds  10254
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